Modeling of cardiac muscle thin films: Pre-stretch, passive and active behavior

CitationShim, Jongmin, Anna Grosberg, Janna C. Nawroth, Kevin Kit Parker, and Katia Bertoldi. 2012. “Modeling of Cardiac Muscle Thin Films: Pre-Stretch, Passive and Active Behavior.” Journal of Biomechanics 45 (5) (March): 832–841.

Published Versiondoi:10.1016/j.jbiomech.2011.11.024

Permanent linkhttp://nrs.harvard.edu/urn-3:HUL.InstRepos:11949243

Terms of UseThis article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP

Share Your StoryThe Harvard community has made this article openly available.Please share how this access benefits you. Submit a story .

Accessibility

Supporting Material for

Modeling of Cardiac Muscular Thin Films:

Pre-stretch, Passive and Active Behavior

Jongmin Shim1, Anna Grosberg2, Janna C. Nawroth3,Kevin Kit Parker2, and Katia Bertoldi1

1School of Engineering and Applied Science, Harvard University, Cambridge, MA2Disease Biophysics Group, Wyss Institute for Biologically Inspired Engineering, School of Engineering

and Applied Science, Harvard University, Cambridge, MA3Division of Biology, California Institute of Technology, Pasadena, CA

S1 EXPERIMENTAL METHODS S1

S1 Experimental Methods

This section provides a detailed description of the preparation of the silicone thin film and cellharvesting procedure.

S1.1 Manufacturing the Silicone Thin Films

The silicone thin films were prepared via a multi-step spin coating process, extending a previouslypublished method (Feinberg et al., 2007) in order to allow for higher experimental throughput.The samples prepared this way have been dubbed “Heart on a chip”. To prepare the chips, asection of glass (7.5cm× 11cm) was sonicated and covered with a protective film (Static ClingFilm, McMaster-Carr, Robbinsville, NJ) on both sides. Rectangles (1cm × 0.6cm) were cutout of the top protective film, and poly(N-isopropylacrylamide) (PIPAAm, Polysciences, Inc.,Warrington, PA) in 1-butanol (10% w/v) was spin-coated onto the uncovered portions of theglass, and then the top protective film was removed. The silicone polymer polydimethylsiloxane(PDMS, Sylgard 184-Dow Corning, Midland, MI) was pre-cured for 1 to 6 hours, spin coatedonto the glass, and cured for 8 hours at 65◦C. The bottom protective film was removed, and theglass section was cut into cover slips of 1.4cm× 1.2cm. From each glass section, two cover slipswere used to measure the thickness of the PDMS layer using a profilometer (Dektak 6M, VeecoInstruments Inc., Plainview, NY). The PDMS surface of the cover slip was functionalized andsterilized for 8 minutes in UV ozone (Model No. 342, Jetlight Company, Inc., Phoenix, AZ).Fibronectin (FN, Sigma, St. Louis, MO or BD Biosciences, Sparks, MD) was micropatternedonto the surface via microcontact printing with a brick pattered PDMS stamp (each brick was20µm wide and 100µm long, and each short edge terminated with 5µm long saw-tooth, Brayet al., 2008), such that the brick pattern would be perpendicular or at some tilted angle to thelong edge of the cover slip. The gaps in the pattern were blocked by incubating the cover slip inPluronics F127 (BASF Group, Parsippany, NJ) for 5 minutes, and then the chips were storedat 4◦C until cell seeding. Three different PDMS thickness were made though this procedure:14.5µm, 18.0µm, 23.0µm.

S1.2 Cell harvesting, Seeding, and Culture

Cell harvesting was conducted in accordance with the guidelines of Institutional Animal Careand Use Committee of Harvard University. Cardiomyocytes were harvested and isolated basedon published protocols (Adams et al., 2007). Briefly, ventricles were extracted from two-day-oldneonatal Sprague-Dawley rats (Charles River Laboratories, Wilmington, MA) and homogenizedby washing in Hanks balanced salt solution, then incubated with trypsin overnight at 4◦C. Torelease the myocytes into solution, the ventricular tissue was digested with collagenase at 37◦C.Cells were re-suspended in M199 culture medium supplemented with 10% (v/v) heat-inactivatedfetal bovine serum (FBS), 10mM HEPES, 3.5g/L glucose, 2mM L-glutamine, 2mg/L vitaminB-12, and 50U/ml penicillin. The myocytes were seeded on the substrates at a density of 400,000cells per cover slip (round, diameter of 18mm), and cultured for a period of four days prior torunning the assay. During the initial 48 hours, the cells were kept at 37◦C and 5%CO2 with10% FBS media in the culture; thereafter, the maintenance media (2% FBS) was used. Themeasured cell thickness was approximately 4.0µm.

S2 CONSTITUTIVE MODELING OF BIO-HYBRID FILMS S2

S1.3 Contractility Experiments

To run the contractility experiments, the chip was moved to a Petri dish with 37◦C normalTyrode’s solution (1.192g of HEPES, 0.901g of glucose, 0.265g of CaCl2, 0.203g of MgCl2,0.403g of KCl, 7.889g of NaCl, and 0.040g of NaH2PO4 per liter of deionized water; reagentsfrom Sigma, St. Louis, MO). The PDMS elastic layer was cut such as to leave the PDMSalong the border of the cover slip intact, while in the center there were two rows of three tofour rectangles (approximately 2mm × 3.5mm) separated by approximately 1mm, and con-nected to the PDMS border by one edge (see Figs. ??). The unwanted film was peeled away,exposing the temperature-sensitive PIPAAm beneath the rectangular strips and causing it todissolve as the solution was allowed to cool down towards room temperature. Subsequently,the rectangular films were free to move vertically up from the plane of the cell culture, withoutcompletely separating from the chip. The chip with six or eight rectangular films was moved toa heated bath (34◦C − 37◦C) under a stereomicroscope (Model MZ6 with darkfield base, LeicaMicrosystems, Inc., Wetzlar, Germany), and the dynamics of the films was recorded from aboveat 120 frames/sec ( Basler A602f camera, Exton, PA). Several snapshots from the recordedmovies are shown in Figs. ?? and ??.

While the film laid flat on the glass, both its length and width were measured. As the filmscontracted away from the surface, the horizontal projection was recorded. Unimpeded filmswould first isometrically deform to a curvature corresponding to the diastolic tension in thecells, and then continue to deform until peak systole. For substrates with cells aligned with thelength of the film, the observed deformation was a shortening of the projection, described inFig. ?? (top). On the other hand, the substrates with cells at an angle to the length of the filmshowed the inclined free edge of the film (see Fig. ?? (bottom)), so an appropriate projectionlength needed to be defined in order to obtain the curvature of the MTFs. To ensure all films onthe chip were moving at once, the myocytes were field-stimulated at 1.5Hz with 5 to 20V usingan external field stimulator (Myopacer, IonOptix Corp., Milton, MA) which applied a 10msecsquare wave pulse.

S2 Constitutive Modeling of Bio-hybrid Films

In this section, we first summarize the equations governing the nonlinear deformation of hyper-elastic materials, following the formulation previously introduced by ?. Then, the equations arespecialized to capture the behavior of the elastomeric substrate and the cardiac myocytes.

As mentioned in Section 1, cardiac myocytes comprise highly aligned myofibrils, so theirbehavior is captured extending the formulation previously developed to model fiber reinforcedelastomers (e.g., Spencer, 1984).

S2.1 General Formulation

Let F = ∂x/∂X be the deformation gradient mapping a material point from the reference po-sition X to its current position x, and J = detF be its determinant. The behavior of nearlyincompressible materials is effectively described by splitting the deformation locally into volu-metric (denoted by superscript v) and isochoric (denoted by superscript i) components as

F = Fv · Fi, where Fv = J1/31, Fi = J−1/3F. (S1)

S2 CONSTITUTIVE MODELING OF BIO-HYBRID FILMS S3

For an isotropic hyperelastic material, the strain energy density ψ is a function of the threeinvariants of C = FT ·F, ψ = ψ(I1, I2, I3). Based on the kinematic assumption (S1), a decoupledform for ψ is introduced

ψ = ψv(J) + ψi(I1, I2), (S2)

with

I1 = trCi, I2 =1

2

[(trCi

)2 − tr(Ci2)]. (S3)

When an isotropic material is reinforced by a family of fibers with direction a0 in the referenceconfiguration, the isochoric part of the strain energy can be expressed as a function of not onlyI1 and I2, but also two additional invariants (I4, I5) depending on a0 (Spencer, 1984),

ψi = ψi(I1, I2, I4, I5), (S4)

withI4 = a0 ·Ci · a0 = λ2, I5 = a0 ·Ci2 · a0. (S5)

However, the dependence of the material response on I2 or I5 is generally weak, so that theisochoric part of the strain energy can be simplified as

ψi = ψiiso(I1

)+ ψiani

(I4

), (S6)

where ψi has been further decoupled into an isotropic, ψiiso, and anisotropic, ψiani, contributions.Finally, the Cauchy stress T is found by differentiating ψ with respect to C as

T =2

JF∂ψ

∂CFT =

∂ψv

∂J1 +

2

J

[∂ψiiso

(I1

)∂I1

dev(Bi)

+ I4∂ψiani

(I4

)∂I4

dev (a⊗ a)

], (S7)

where “dev” stands for deviatoric part of 2nd order tensors, Bi = Fi·FiT is the left Cauchy-Greentensor, and a = Fi ·a0/

√I4 is the unit directional vector of fiber in the deformed configuration.

In the following two sections, the above general formulation is specialized to capture thebehavior of the elastomeric substrate and the cardiac myocytes.

S2.2 Elastomeric Substrate Behavior

The elastomeric substrate is fabricated using PDMS, and its behavior is well captured using aneo-Hookean model, so that the strain energy density is given by

ψ = ψv (J) + ψi(I1

)=κ

2(J − 1)2 +

E

6

(I1 − 3

), (S8)

where κ and E denote the bulk modulus and the initial elastic modulus of the elastomer,respectively.

Based on (S7), the Cauchy stress is given by:

T = κ(J − 1)1 +E

3Jdev

(Bi). (S9)

S2 CONSTITUTIVE MODELING OF BIO-HYBRID FILMS S4

S2.3 Cardiac Muscle Cell Behavior

The characteristic energy dissipative process of the cardiomyocite active contribution can bemodeled by the physiological approach, which considers its true mechanism due to the actinpotential under calcium flux (e.g., Sainte-Marie et al., 2006; Chapelle et al., 2010). However,this approach is mathematically complex, and model parameter identification is challenging. Inthis article, instead, we take a simple, but effective phenomenological approach, where the activebehavior is assumed to be elastic. While recently Bol et al. (2009) developed a phenomenologicalmodel of cardiac muscle cells, their model neglects the important effect of the pre-stretch thatthe muscle cells develop as they mature. This paper presents a 3-D phenomenological model thatcaptures the two major features of cardiac muscles: the passive behavior inducing pre-stretcheddeformation and their active behavior including systole and diastole.

S2.3.1 Kinematics including Pre-stretch

Even when resting, muscle cells in vivo experience a state of stress due to their pre-stretchedconditions. This can be clearly observed from the experimental data reported in Figs. ?? andFig. ??, showing a non-negligible MTF curvature during diastole. In order to account forsuch pre-stretched conditions, previous constitutive models introduced an ad hoc strain-shiftin the stress-strain curve (Blemker et al., 2005; Bol and Reese, 2008; Calvo et al. (2010)).However, this strain-shift leads to a stress-strain curve that depends on the level of pre-stretch.In this paper, a different approach is adopted; inspired by the multiplicative decompositionintroduced by Kroner-Lee (Kroner, 1960; Lee, 1969), the isochoric deformation gradient Fi in(S1) is decomposed into load-induced, FiL, and pre-stretched, FiS , contributions

Fi = FiL · FiS . (S10)

Here, for the sake of simplicity, the pre-stretch is assumed to be fully developed during celldifferentiation and maturation prior to the experiment, and not to be affected by the cell activeresponse. If we also assume that the pre-stretch deformation is incompressible and the cellsdeform affinely, the pre-stretched contribution to the deformation gradient FiS is given by

FiS = QΛQT , (S11)

where

Λ = λS e1 ⊗ e1 +(λS

)− 12

(e2 ⊗ e2 + e3 ⊗ e3) (S12)

Q = cos θ (e1 ⊗ e1 + e2 ⊗ e2) + sin θ (−e1 ⊗ e2 + e2 ⊗ e1) + e3 ⊗ e3 (S13)

where λS is the pre-stretch induced into the cardiac myocytes lying in the x1-x2 plane duringmaturation (i.e., myogenesis and tissue development) and θ is the angle identifying the cellalignment in the undeformed configuration (see Fig. ??). The diastole curvatures of the exper-imental data reported in Figs. ?? and Fig. ?? clearly show a deviation in the pre-stretchedresponse of the cells as a result of the diversity in cell conditions. Therefore, we expect theparameter λS not to be uniquely determined, but to be influenced by experimental factors suchas ambient temperature/humidity, and age/health of the cells.

S2 CONSTITUTIVE MODELING OF BIO-HYBRID FILMS S5

S2.3.2 Constitutive Equations for Passive and Active Behavior

In order to describe both passive (representing resting status) and active (including systole anddiastole) behavior of cardiomyocytes, we specify their free energy as

ψ = ψv (J) + ψiiso(I1

)+ ψipani

(I4

)+ ψiaani

(I4, q

), (S14)

where q is the activation level of cardiac muscle cells. In the present model, while ψv and ψiisoreflects the isotropic contribution of the intercellular part, ψipani and ψiaani represent the passiveand the active contributions of anisotropic effect of the myofibril, repectively.

Here, as for the elastomeric substrate, the volumetric free energy term is chosen as

ψv(J) =κ

2(J − 1)2 , (S15)

and the isotropic contribution of the intercellular part is captured using a neo-Hookean formu-lation

ψiiso =Ec6

(I1 − 3

), (S16)

where κ denotes the bulk modulus of the cells and Ec is the initial elastic modulus of intercellularpart. Several experimental studies (e.g., Le Guennec et al., 1990; Granzier and Irving, 1995;Palmer et al., 1996; Cazorla et al., 2000; Weiwad et al., 2000; Boron and Boulpaep, 2009) revealthat the passive response of cardiac cells is characterized by an exponential relation betweenstress and deformation (see Fig. ??). In order to capture such behavior of cardiac cells, wepropose the following free energy

ψipani(λ)

=Epα2

[eα(λ−1) − α(λ− 1)− 1

], (S17)

where λ =√I4, Ep is the initial elastic modulus, and α characterizes the monotonically in-

creasing slope of the stress-strain relation of the passive response of the fibers. For the activeresponse (see Fig. ??), a bell-type relation between stress and deformation has been shown byseveral studies (e.g., Zajac, 1989; Weiwad et al., 2000; Boron and Boulpaep, 2009). Here, wesimplify the active stress-stretch relation into a quadratic form, which is captured introducingthe following free energy

ψiaani(λ, q)

=

P q

2(1−λo)2

[2(1− 2λo) ln λ+ (4λo − λ− 1)(λ− 1)

], if 1 < λ < (2λo − 1)

0, otherwise(S18)

where P is the isometric twitch stress (i.e., the maximum contraction force per unit cross-sectional area) and λo is the optimal stretch (i.e., the stretch at which the maximum activestress is attained). Since Fig. ?? and Fig. ?? clearly show that diversity in cell conditionscauses large variations of the peak amplitudes, as the pre-stretch λS , the parameter P is alsotreated as a non-uniquely determined property of cultured cardiac tissue. Finally, the activationlevel q is a function of time and is defined as

q(t) =

(t

T

)2

exp

[1−

(t

T

)2], (S19)

S3 PARAMETER IDENTIFICATION FOR THE CONSTITUTIVE MODEL S6

where T is the characteristic contraction time.Based on (S7), the proposed free energy results in the following Cauchy stress

T = Tv + Tiiso + Tip

ani + Tiaani, (S20)

where

Tv = κ(J − 1)1 (S21)

Tiiso =

Ec3J

dev(Bi)

(S22)

Tipani =

Epλ

αJ

[eα(λ−1) − 1

]dev(a⊗ a), (S23)

Tiaani =

P qJ

[1−

(λ−λo1−λo

)2]

dev(a⊗ a), if 1 < λ < (2λo − 1)

0, otherwise(S24)

S3 Parameter Identification for the Constitutive Model

This section presents the procedure to identify the 11 material parameters entering into theproposed 3-D constitutive model. While the material parameters for the elastomeric substrateare provided by the manufacturers, the parameters needed for the cardiac muscle cells areobtained from both literature and contraction assays performed with neonatal rat ventricularmyocytes micropatterned on PDMS thin films.

S3.1 Summary of Model Parameters

The proposed constitutive model requires 11 material constants

• Elastomeric substrateE Initial elastic modulusκ Initial bulk modulus

• Pre-stretched deformation of cardiac muscle cellsθ Angle defining the initial alignment of the cells within the x1 − x2 planeλS Pre-stretch developed in the cardiomyocytes during cell maturation

• Volumetric and isotropic behavior of cardiac muscle cellsEc Initial elastic modulus of intercellular partκ Initial bulk modulus

• Passive fiber behavior of cardiac muscle cellsEp Initial elastic modulus of fiberα Constant related to the slope of stress-strain relation of passive fiber

• Active fiber behavior of cardiac muscle cellsP Isometric twitch stressλo Optimal stretch for the maximum active stressT Characteristic time scale for contraction

S3 PARAMETER IDENTIFICATION FOR THE CONSTITUTIVE MODEL S7

S3.2 Elastomeric Substrate (E, κ)

Only two material constants (E, κ) are required to model the elastomeric substrates. Theinitial elastic modulus is directly specified by the manufacturer as E = 1.5 MPa, while itsincompressible behavior (ν = 0.49) yields an initial bulk modulus κ = 25 MPa.

S3.3 Cardiac Myocytes

The proposed model requires seven material-specific parameters (κ, Ec, Ep, α, P , λo, T ) and two

geometric-specific parameters (θ, λS ). Most of model parameters are available in the literature.The paper by Weiwad et al. (2000) provides in a single article nearly all the required test resultsto identify the material paramors entering into the proposed material model. Moreover, all theirdata are in a good agreement with similar test results for the heart muscle of adult rats (i.e.,Granzier and Irving, 1995, Palmer et al., 1996 for passive behavior and Nishimura et al., 2000,Palmer et al., 1996 for active behavior). Thus, the test results presented in Weiwad et al. (2000)are used to identify five material-specific parameters entering into the proposed constitutivemodel.

Experiments with straight cell alignment (Section 2) are used to identify the remaining threeparameters; two of which (i.e., P , λS) are experimental-conditions-specific, and one of which(i.e., T ) is the characteristic time scale of contractile behavior under the stimulating electricpulse. Here, we assume that the cell conditions affect only the twitch stress P and the pre-stretch λS .

S3.3.1 Isochoric Behavior of Intercellular Part (Ec) and Isochoric Passive Behaviorof Fiber Part (Ep, α)

Several researchers have reported experimentally measured values for Young’s modulus of non-activated rat cardiac myocytes by using atomic force microscopy (e.g., Lieber et al., 2004) orcustom-designed force transducer (e.g., Weiwad et al., 2000). The measured values varies from20kPa to 45kPa depending on the experimental protocols and the ages of rats. FollowingWeiwad et al. (2000), we use a Young’s modulus, Epassive = 23kPa (see Fig. ??). From (S20),recall that the measured Young’s modulus of the non-activated cell can be decomposed into thecontribution of intercellular part and passive fiber part. Here, we assume that the passive fiberpart is mainly responsible for the passive elastic behavior, so that Ep = 21kPa and Ec = 2.3kPa.

Note that the passive stress (T passive11 ) can be expressed as:

T passive11 =(T iiso

)11

+(T ipani

)11, (S25)

where using incompressibility each stress contribution can be determined from the reduced formof (S22) and (S23):

(T iiso

)11

=Ec3

(λ2 − 1

λ

), (S26)(

T ipani

)11

=Epλ

α

[eα(λ−1) − 1

]. (S27)

By curve-fitting the experimental results with the above equations (see Fig. ??), we obtainα = 5.5.

S4 PARAMETRIC STUDIES S8

S3.3.2 Isochoric Active Behavior of Fiber Part (P , λo, T )

Weiwad et al. (2000) have reported the evaluation of the isometric twitch stress as a function ofcalcium concentration (see Fig. ??). Note that a constant calcium concentrations of pCa = 2.74was employed throughout the essay tests presented in Section 2. In order to get its overalldependence on the stretch, all the active tension-stretch relations are normalized by its maximumvalue, and then curve-fitted by the proposed quadratic form of active stress-stretch relation foruniaxial loading case:

(T iaani

)11

=

P q[1−

(λ−λo1−λo

)2], if 1 < λ < (2λo − 1)

0, otherwise(S28)

As a result, we obtain λo = 1.24, as shown in Fig. ??. As described in Section 4, the range ofthe isometric twitch stress of P ∈ [2.8, 21.6]kPa is determined from experiments with straightcell alignment presented in Section 2. The characteristic contraction time can be determined bymeasuring the time when the signal reaches its maximum value after the activation starts. Fig.?? shows a typical profile of the active amplitude history, and T = 0.21sec was determined.

S3.3.3 Volumetric Behavior (κ)

For simplicity, we assume that the cardiac muscle is nearly incompressible and its volumetricbehavior is determined by the passive response. Thus, using Epassive = 23kPa and ν = 0.49, we

obtain a bulk modulus of κ =Epassive

3(1−2ν) = 380kPa.

S3.3.4 Isochoric Pre-stretch Behavior (θ, λS)

For the model parameter identification, the controlled test set (i.e., the case with the straight cellalignment) with θ = 0◦ is used. It is known that the level of pre-stretch (or pre-stress/pre-load)developed during cell maturation depends on the properties of substrate.

From the assay experiments in Section 2, we obtain λS ∈ [1.11, 1.18]. This range of valuesfor pre-stretch have been also reported under similar experimental conditions using a siliconsubstrate (e.g., Mansour et al., 2004).

S4 Parametric Studies

In this Section, we explore the effect of important parameters within MTF such as the PDMSthickness, the isometric twitch stress of cells, the pre-stretch of cells, and the frequency of field-stimulation.

S4.1 Effect of PDMS Thickness

Here, the effect of PDMS thickness on the response of the MTF is explored by varying t from13.0µm to 28.0µm. Meanwhile, the average value of the two cell-condition-dependent parametersare used (i.e., λS = 1.14 and P = 9.0kPa), and all other model parameters are also unchangedthroughout the simulations.

Fig. S1a shows the effect of PDMS thickness on the total curvature (K) of MTF. As expected,an increase in PDMS thickness leads to a decrease in curvature. To quantify this behavior, the

S4 PARAMETRIC STUDIES S9

total curvature is decomposed into pre-stretch (Kp), and active (Ka), contributions, so that

K = Kp +Ka. (S29)

Fig. S1b reports the evolution of maximum passive and active curvature contributions (i.e.,max(Kp) and max(Ka)) as a function of the PDMS thickness. The pre-stretch contribution ofthe curvature can be also estimated by applying a recent modification of the Stoney’s equation(i.e., ?). Assuming that the layer of cells is much thinner than that of substrate, the Stoneycurvature equation predicts the curvature of the bilayer film as

Kstoney =6(Ec + Ep) lnλS

Et

(t

t

)(1 +

t

t

). (S30)

The pre-stretch induced curvature predicted by (S30) is also reported in Fig. S1b, and a goodagreement with the pre-stretch contribution from the FEM calculations is observed.

S4.2 Effect of Isometric Twitch Stress of Cells

We vary P from 0 to 30.0kPa while keeping all other parameters unchanged, including t =18.0µm and λS = 1.14.

Fig. S1c shows the effect of isometric twitch stress on the curvature of MTFs. While thepassive contribution of the curvature (max(Kp)) is unaffected by P , the active contribution(max(Ka)) is observed to vary almost linearly with P (Fig. S1d).

S4.3 Effect of Cell Pre-stretch

The effect of pre-stretch on the cardiac muscle cell response is investigated by varying λS from1.00 (i.e., no pre-stretch) to 1.39 while keeping all other parameters unchanged and using t =18.0µm and P = 9.0kPa.

Figure S1e reports the time history of film curvature for various level of cells pre-stretch.The results clearly show that the level of cell pre-stretch affects the curvature both duringcell maturation and activation. Interestingly, for the active response, we observe the quadraticrelation between the pre-stretch and the active curvature contribution. Thus, this study suggeststhat the maximum level of active response (i.e., the maximum active mobility) for the MTFscan be achieved by introducing an optimal pre-stretch value. Moreover, Fig. S1f clearly showthat the Stoney equation (S30) largely underestimates the MTF curvature for large pre-stretch.

REFERENCES S10

References

W.J. Adams, T. Pong, N.A. Geisse, S.P. Sheehy, B. Diop-Frimpong, and K.K. Parker. Engineer-ing design of a cardiac myocyte. Journal of Computer-Aided Materials Design, 14(1):19–29,2007.

S.S. Blemker, Pinsky P.M., and S.L. Delp. A 3d model of muscle reveals the causes of nonuniformstrains in the biceps brachii. Journal of Biomechanics, 38:657–665, 2005.

M. Bol and S. Reese. Micromechanical modelling of skeletal muscles based on the finite elementmethod. Computer Methods in Biomechanics and Biomedical Engineering, 11:489–504, 2008.

M. Bol, S. Reese, K.K. Parker, and E. Kuhl. Computational modeling of muscular thin film forcardiac repair. Computational Mechanics, 43:535–544, 2009.

W.F. Boron and E.L. Boulpaep. Medical Physiology. Saunders, 2009.

M.A. Bray, S.P. Sheehy, and K.K. Parker. Sarcomere alignment is regulated by myocyte shape.Cell Motility and the Cytoskeleton, 65(8):641–651, 2008.

B. Calvo, A. Ramirez, A. Alonso, J. Grasa, F. Soteras, R. Osta, and M.J. Munoz. Passivenonlinear elastic behaviour of skeletal muscle: Experimental results and model formulation.Journal of Biomechancis, 43:318–325, 2010.

O. Cazorla, J.-Y. Le Guennec, and E. White. Lenght-tension relationships of sub-epicaridal andsub-endocardial single ventricular myocytes from rat and ferret hearts. Journal of Molecularand Cellular Cardiology, 32:735–744, 2000.

D. Chapelle, J.F. Gerbeau, J. Sainte-Marie, and I.E. Vignon-Clementel. A poroelastic modelvalid in large strains with applications to perfusion in cardiac modeling. Computational Me-chanics, 46:91–101, 2010.

A.W. Feinberg, A. Feigel, S.S. Shevkoplyas, S. Sheehy, G.M. Whitesides, and K.K. Parker.Muscular thin films for building actuators and powering devices. Science, 317(5843):1366–1370, 2007.

H.L. Granzier and T.C. Irving. Passive tension in cardiac muscle: contribution of collagen, titin,microtubules, and intermediate filaments. Biophysical Journal, 68:1027–1044, 1995.

E. Kroner. Allgemeine kontinuums theorie der versetzungen und eigenspannungen. Archive forRational Mechanics and Analysis, 4:273–334, 1960.

J.-Y. Le Guennec, N. Peineau, J.A. Argibay, K.G. Mongo, and D. Garnier. A new methodof attachment of isolated mamalian ventricular myocytes for tension recording: Length de-pendence of passive and active tension. Journal of Molecular and Cellular Cardiology, 22:1083–1093, 1990.

E.H. Lee. Elastic-plastic deformation at finite strains. Journal of Applied Mechanics, 36:1, 1969.

S.C. Lieber, N. Aubry, J. Pain, G. Diaz, S.J. Kim, and S.F. Vatner. Aging increases stiffnessof cardiac myocytes measured by atomic force microscopy nanoindentation. AJP-Heart andCirculatory Physiology, 287:H645–H651, 2004.

REFERENCES S11

H. Mansour, P.P. de Tombe, A.M. Samarel, and B. Russell. Restoration of resting sarcomerelength after uniaxial static strain is regulated by protein kinase cε and focal adhesion kinase.Circulation Research, 94:642–649, 2004.

S. Nishimura, S. Yasuda, M. Katoh, K.P. Yamada, H. Yamashita, Y. Saeki, K. Sunagawa, R. Na-gai, T. Hisada, and S. Sugiura. Single cell mechanics of rat cardiomyocytes under isometric,unloaded, and physiologically loaded conditions. AJP-Heart and Circulatory Physiology, 287:H196–H202, 2000.

R.E. Palmer, A.J. Brady, and K.P. Roos. Mechanical measurements from isolated cardiacmyocytes using a pipette attachment system. AJP-Cell Physiology, 270:C697–C704, 1996.

J. Sainte-Marie, D. Chapelle, R. Cimrman, and M. Sorine. Modeling and estimation of thecardiac electromechanical activity. Computers and Structures, 84:1743–1759, 2006.

A.J.M. Spencer. Continuum Theory of the Mechanics of Fibre-reinforced Composites. Springer-Verlag, Wien-New York, 1984.

W.K.K. Weiwad, W.A. Linke, and M.H.P. Wussling. Sarcomere length-tension relationship ofrat cardiac myocytes at lengths greater than optimum. Journal of Molecular and CellularCardiology, 32:247–259, 2000.

F.E. Zajac. Muscle and tendon: Properties, models, scaling, and application to biomechanicsand motor control. Critical Reviews in Biomedical Engineering, 17:359–411, 1989.

REFERENCES S12

(a) (b)

(c) (d)

(e) (f)

Figure S1: The left column reports the time-curvature plots from FE analysis while the rightcolumn the stress contribution of pre-stretch and active response (max(Sp) and max(Sa)). (a,b) effect of PDMS thickness, (c, d) Effect of isometric twitch stress of cells, (e, f) effect of cellpre-stretch.